A Swift Introduction to Geometric Algebra

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3 жыл бұрын

This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.
This video was made using manim (github.com/ManimCommunity/manim), a math animation library.
Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.
Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the outer product of vectors. A k-blade is a blade of grade k. Some of the terminology between linear algebra and geometric algebra can be confusing, such as the use of the terms vector space and dimension. Because multivectors form their own vector space, but we don't consider all of them to be vectors, some people use the term linear space instead of vector space. While we think of dimension in terms of spatial degrees of freedom, the mathematical definition means that the 3D geometric algebra shown here is actually eight-dimensional. This is part of the reason for the distinction between the terms grade and dimension.
The following are the things that I purposefully got wrong to make things simpler:
The biggest thing I glossed over is the distinction between blades and vectors. A lot of the ways that I described k-vectors intuitively actually only applied to k-blades, not k-vectors. For example, in four or more dimensions, all 2-blades can be represented as an oriented area, but not all 2-vectors can. I didn't mention this fact for two reasons: first, for a first look, the distinction is not that important. Second, the distinction doesn't even matter until you reach four or more dimensions. In three dimensions, all k-vectors are k-blades.
My use of the term basis was not quite right, and I should have used the term "orthonormal basis". I wanted this video to be understood by people who had learned about vectors in a physics class where they don't go into too much detail about the exact definitions a linear algebra class would give you, so I didn't want to use the term orthonormal. I tried to mitigate this somewhat by always using terms like "the basis" and assuming that it was known that I meant the standard orthonormal basis. Hopefully this doesn't cause any confusion.
I mentioned that the inner product can get complicated when generalized to generic blades. What I didn't mention is that there is a large amount of disagreement on what this generalization is. I've seen at least five different extensions of the inner product to higher-grade blades. I didn't mention this because it was not pertinent to the discussion. However, even though I didn't talk much about the inner and outer products in this video, they are essential to the usage of geometric algebra in many applications.
This wasn't necessarily wrong, but I assumed that space is Euclidean throughout the video. In some applications, especially in relativity, space is not Euclidean, and a few things aren't the same.
In general, the pseudoscalar for a geometric algebra is the highest-grade element, but this does not always square to -1. In the three main applications of geometric algebra (2D Euclidean space, 3D Euclidean space, and 4D Minkowski space), this does happen to be the case, but in general it is not true.
When I mentioned dividing vectors, I didn't mention the fact that when dividing, you have to "divide on the left" or "divide on the right" because multiplication is not commutative. I actually prefer just to divide by multiplying by the inverse, because then you don't have to worry about how you're dividing. I didn't mention this because I never mentioned inverses again after I introduced them.
Another issue with inverses is that while vectors have inverses, not all multivectors have inverses. Also, in other spaces, such as the Minkowski space used in relativity, even some vectors don't have inverses. Again, this was because I didn't really use inverses much after this.
This one is small, but we assumed several properties of the geometric product that we didn't prove or state, such as distributivity and associativity. To be precise, the set of multivectors along with addition and the geometric product forms an associative algebra. I defined the geometric product in a bit of a roundabout way and I couldn't find a good place to insert these facts. The kind of people this video is aiming at are those who don't know a lot of abstract algebra and so I assumed that they would assume associativity and distributivity.
Rotors actually have many different definitions, so don't always think of them as a complex exponential.

Пікірлер: 1 500

@sudgylacmoe 3 жыл бұрын

I just realized a stupid mistake I made: at 8:20, this should actually be that the inner product is this length times ||u||. I don't know how I didn't notice this in all of my times watching this video while making it. EDIT: There is now an addendum to this video here: kzfaq.info/get/bejne/Zsh_nNxhrbyxoXU.html. It clears up a few misleading things in this video and then focuses on actually understanding the geometric product. EDIT 2: Another correction: at 29:00, the equation should be i a · (b × c) = a ∧ b ∧ c, not a · (b × c) = i a ∧ b ∧ c.

@daemond8093 3 жыл бұрын

No worries You got your point through...that is more than enough, Thanks a lot for such an explanation. I want to apply geometric algebra to viscoelastic equations. Whenever viscosity is introduced in any media, it is associated with a complex number and this results in the rotation of the velocity vector. Please let me know if you are interested in Collaborating.

@yabgu79 3 жыл бұрын

I wish I could upvote thousand of times. You are a hero thank you for the great work. I have tried to understand this for like 10 years and I always thought 3d rotation is a magic box just copy&paste and it works, however, it works don't even try to think about it but after seeing rotors as bivectors I can understand what all these were about. Thank you a lot keep it up this is the only channel I have clicked the bell icon,

@steveoh907 3 жыл бұрын

@@daemond8093 I would like any public updates from your collaboration with sudgylacmoe....if that’s possible 👍

@3metrictensor 3 жыл бұрын

I made exactly that same mistake once while I was showing it to someone. I think it is because so often you are used to working with an orthonormal basis that you go into auto pilot.

@truthphilic7938 2 жыл бұрын

can make a video about "pseudovector and pseudoscalar"?

@ConnorMcCormick 2 жыл бұрын

Every "wait a minute" I have to pause to wipe tears from my eyes so I can see the screen

@drdca8263 2 жыл бұрын

Does doing this happen to take about a minute?

@_wetwillyinc 2 жыл бұрын

"wait a minute" >takes a shot

@nzuckman 2 жыл бұрын

All I think of is Kazoo Kid

@affablegiraffable 2 жыл бұрын

This video is like deeply unsettling

@gabitheancient7664 2 жыл бұрын

SAME

@mysteriousgrimreaper7749 5 ай бұрын

There’s something really beautiful about the fact that simply choosing to ignore “you can’t multiply vectors” leads to ALL OF PHYSICS

@adixo1851 2 жыл бұрын

Imaginary Numbers being Pseudo-Scalars is better plot twist then some movies 😂

@neilfmoore 9 ай бұрын

The fact that geometric algebra and algebraic geometry are completely different things just proves that mathematical language is non-Abelian.

@hassanchehime5745 28 күн бұрын

Very😂

@appybane8481 16 күн бұрын

I think it's just not commutative like those multivector in geometric algebra

@mrpedrobraga 3 жыл бұрын

This sounds like initiation in a cult, and i'm all for it.

@Utesfan100 2 жыл бұрын

To become truly initiated you must learn the octonions.

@mzg147 2 жыл бұрын

Many mathematicians see geometric algebra as a cult, so you are right :D

@robfielding8566 2 жыл бұрын

@@Utesfan100 actually.... at 3D, it's not really quaternions. Quaternions are missing the 3D 1D vectors and pseudo-scalar. Octonions are a different direction entirely; a wrong-turn IMHO. As you go into higher dimensions with GA, octonions are not a thing you run into. Octonions are not associative; etc. I think the O(2^n) growth of the calculations as dimensions go up is why we did not go this direction when it was discovered. It's almost too tedious without computer assistance.

@HClutchwao 2 жыл бұрын

Lol - good initiation. I am all for it as well.

@Utesfan100 2 жыл бұрын

@@robfielding8566 I was merely attempting to point out that if you think geometric algebra has a cult following, you should look to the composition algebras :) As a devotee, I am compelled to address your comments more fully. I concede that the ability of vector notation to scale indefinitely lead to its wide adoption across all of mathematics. Which is a more fitting property for a geometric algebra, associativity or compatibility of the product with length? This latter property is what defines a composition algebra. Merab Gogberashvili has done some interesting things using the split-octonions in place of Cl(0,3), analogous to paravectors. This algebra is the unique real composition algebra that contains Minkowski 3+1 space. (There is even a Cauchy integral formula for this algebra.) The relationship between the octonions and the exceptional Lie groups is deep. If one encounters the latter, they would be well served to understand the former. Speaking of cult followings, E_8 comes to mind :) While not a Clifford algebra, the composition property gives the octonions a very geometric flavor. Indeed, Advances in Applied Clifford Algebras frequently includes articles dedicated to the octonions, suggesting they are at least in the same family of algebras. Even if just as the crazy, eccentric uncle.

@618361 2 жыл бұрын

I work in applied electromagnetism and never learned so much in 45 min. You've made sense out of concepts that were always just out of reach. You've changed my life

@julesk1088 Жыл бұрын

Share it with others at work

@curiousaboutscience Жыл бұрын

Right?!

@louieburnham8090 9 ай бұрын

There’s a lot of great literature about this field and it’s applications. Check out Geometric Algebra for Electrical Engineers or Geometric Algebra for Physicists.

@IanBLacy 9 ай бұрын

WHAT THE FUCK IS APPLIED ELECTROMAGNETISM IM STUDYING ELECTRICAL ENGINEERING AND ID BETTER NOT HAVE TO LEARN THIS TO UNDERSTAND MAXWELL

@hugocardenasescoto1314 4 ай бұрын

@@IanBLacy Applied Electromagnetism is probably in reference to transmission line theory and similar radio applications. You don't need it to understand Maxwell, in fact it's in reverse. You need Maxwell to understand transmission line theory and radio frequency circuits. However, you might not see it depending on your undergraduate curriculum.

@gabitheancient7664 2 жыл бұрын

geometric algebra feels like something that was missing from mathematics, it kinda explain things that were weird but proven, now everything makes sense

@VeteranVandal 3 ай бұрын

From as far as I got... It does have some interesting things to it. It does make for a more compact approach too. The only thing I'm worried about is dimensionality of the entities being geometrically produced(?). There are quite a few things in physics that have weird behavior because the way we handle them is mathematically strange, but you get used to it. I knew I could use the external product instead of the cross product and get much the same things out, but the geometric product solves some notation problems we noticed but had to fix afterwards.

@debbiegilmour6171 6 ай бұрын

As soon as you brought up "i", I thought "he hasn't..." but as the vectors were rotated through 90° I was like "he is!!!" then you dropped the big reveal and my face was all Ö That was stunning and brave! Bravo!

@mjjohnson2755 3 жыл бұрын

I don’t normally comment on KZfaq, but as someone trying to understand higher level math concepts, this is one of the best videos for education I’ve ever seen, at least for the way my brain works; thank you so much!

@olegt962 2 жыл бұрын

@@hyperduality2838 who are u

@frankdimeglio8216 2 жыл бұрын

@@olegt962 E=MC2 IS F=ma, as time is NECESSARILY possible/potential AND actual IN BALANCE; AS ELECTROMAGNETISM/energy is gravity. This explains the term c4 from Einstein's field equations. Time dilation ultimately proves ON BALANCE that E=MC2 IS F=ma, AS ELECTROMAGNETISM/ENERGY IS GRAVITY. Gravity IS ELECTROMAGNETISM/energy. Balance and completeness go hand in hand. It all CLEARLY makes perfect sense. By Frank DiMeglio

@cht5086 2 жыл бұрын

If you want to understand a higher level of mathematics you need to study the book of “advanced engineering mathematics” That is about vectors in 2d, 3d; double and triple integrals, surface area, Curve, Green’s theorem, Stokes theorem, surface integrals,, Gradient, differentiate equation in different orders....that would be higher levels. Good luck with your study 📚

@prodbytukoo Жыл бұрын

@@cht5086 I think he meant higher level as more abstract

@hedgefondshenning2346 Жыл бұрын

I can recommend you 3blue1brown. His vids are even better.

@stephendonovan9084 2 жыл бұрын

Sudgy: *casually starts using tau Me: wait a minute

@rescueafterhalfanddoubledo5175 3 жыл бұрын

Middle school teacher: you can’t add scalar and vectors Mathematicians: why the hell not

@JimSmithInChiapas 3 жыл бұрын

Middle-school teachers aren't the only ones who have criticized the "addition" of scalars and vectors -- it's also been criticized by physicists. David Hestenes has given good explanations (summarized below) of what's involved, but he's also given flippant responses that cause trouble for us GA proponents. I wish the uploader hadn't indicated that we can "just add" scalars and bivectors. Hestenes' "good" explanation is that Grassman and Clifford (the 19th-Century originators of GA) identified and defined a certain operation by means of which the information expressed individually by scalars and bivectors can be combined in a very useful symbolic form. Grassman and Clifford chose to call that operation "addition" (what else?), and to represent it via the symbol "+". As Hestenes explains in _New Foundations for Classical Mechanics_ (2nd edition, pp. 28-29), Grassman himself had balked at the idea of "adding" scalars and bivectors, but finally recognized (too late in life) that ... _"The absurdity [of "adding" scalars and bivectors] disappears when it is realized that [that operation] can be justified in the abstract "Grassmanian" fashion which [originated with Grassman, and] has become standard mathematical procedure today. All that mathematics really requires is that the indicated relations and operations be well defined and consistently employed."_ Similar comments apply to the "addition" of bivectors. To answer the criticisms directed at such ideas, I made the video *Answering Two Common Objections to Geometric Algebra* , and wrote a document entitled *Making Sense of Bivector Addition* (available at vixra). The video is part of my playlist "Geometric Algebra of Clifford, Grassman, and Hestenes". Such topics are also under discussion currently in the LinkedIn group "Pre-University Geometric Algebra".

@sreedevm1213 3 жыл бұрын

For a mathematician addition is just a binary operation(which is commutative most of the time).In that sense in any given set you can add things.

@davidherrera4837 3 жыл бұрын

Mathematicians regularly add things that one might feel should not be added. One can form what is called "formal linear combinations". My whole comment here is to justify this and give some examples. This happens in the constructions of objects similar to that of geometric algebra: the exterior algebra of a vector space gotten by formal linear combinations of exterior products of vectors: en.wikipedia.org/wiki/Exterior_algebra Note that k-form is by definition an alternating multilinear map that takes k vectors and returns a scalar. This means that adding a k-form and a j-form does not really make sense, but it does algebraically (that is, abstractly) if you form direct sums of the vector spaces of all k-forms and all j-forms. As the comment that Jim mentions, this abstract construction has nice properties (it is not done just for abstraction sake) because the exterior algebra formed by all such linear combinations of k-forms for different values of k is interesting. You can also do something similar (adding things that do not seem should be added) when forming the tensor space: en.wikipedia.org/wiki/Tensor_algebra The tensor product of k vectors in V is technically a vector in the k-fold tensor product of V. However, since there is a natural way to multiply tensor products of different sizes, it is natural to put them all together into a large space. This is what is done above in the exterior algebra. This construction with tensor products is actually useful in physics (if the Grassmannian theory is complicated and seemingly unmotivated). If you have a Hilbert spaces H (think roughly: C^n an n dimensional complex vector space, where the unit vectors give you a state of a particle in quantum mechanics). Then you can form the k-fold tensor product of H with itself k times to get the space characterizing k independent particles. Now, if you want to consider the space of all states where you might not know the number of particles, if the particles are allowed to change, or something than you can consider the common construction in mathematical physics: Fock space which is the tensor algebra for H. Here is a video about it: kzfaq.info/get/bejne/oKenbLp-0MmnmmQ.html Formal Linear combinations / Sums: math.stackexchange.com/questions/1029851/understanding-the-meaning-of-formal-linear-combination-and-tensor-product math.stackexchange.com/questions/996556/why-formal-linear-combination Often the term "formal" is used in math. This means that usually the math written has some meaning, but you are to ignore the technicalities and focus on some other aspect of it. A famous example of this is the dirac delta "function", which is an example of physicists using a formal expression for an integral. kzfaq.info/get/bejne/ibenrK-bu7fbdmw.html kzfaq.info/get/bejne/id5-icZ0u9CRY5c.html The mathematical theory of Distributions was developed to address "functions" like this. Physicists are currently doing something similar with what is called the Feynman path integral, but we are not currently aware if there is a logical justification of it using mathematics... interesting unsolved question. In knot theory, one sometimes does add things that one "probably should not" add. This is a way of turning combinatorial objects into algebraic objects where invariants are sometimes more easily introduced. For instance, knot theorists form linear combinations c_1v_1 + c_2v_2 where c_1, c_2 belong to Z/2Z. That is, c_1 and c_2 are either 0 or 1 and arithmetic is done modulo 2, i.e. bit arithmetic like in a computer. This is done in such a way that you can form linear combinations of literally anything: 1 square + 1 pentagon. If you don't want to worry about a fraction number of cubes (but have multiplicity), then you form linear combinations with scalars in Z (integers). This is done in de Rham cohomology (the d is the exterior derivative of differential forms, related to the exterior product, and the differential \partial gives you the boundary of the region you are integrating over with some signs attached based on orientation) and the extension of Stokes' Theorem for higher dimensions, as generalizations of The Fundamental Theorem of Calculus in1 dimension, Divergence Theorem in 3 dimensions , Green's Theorem in 2 dimensions, Stokes' Theorem in 3 dimensions. Introduction to Knot Theory: kzfaq.info/get/bejne/sLSWmb1kusmlgqc.html Another example of Z-linear combinations is at 57:23 of kzfaq.info/get/bejne/l79ha814ys_RY2w.html If you don't want to worry about negative numbers, you can form linear combinations over Z/2Z as I introduced it above. You can see kzfaq.info/get/bejne/iKeidKtymLXJZ4E.html If you start at 42:20 and go to the end of the presentation and keep an eye out for sums of things that do not have a natural sum, then you can see that it fundamental to the construction. He starts with making some paths into an algebra: so you can multiple by adjoining trees and can add formally. Then he does a lot of algebra involving these trees. He later takes an infinite sum of these graphs wen defining "the differential" At 45:23 the lecturer says "My algebra is all over Z/2Z out of some horrible laziness" referring to the idea that algebra involving scalars in Z/2Z is much, much simpler than over Z since arithmetic only involves 2 numbers, as discussed above.

@blinded6502 3 жыл бұрын

You can't add scalars and vectors. In fact, you can't even add vectors and vectors! Because original definition of addition - is "scalar+scalar", and that's it. Every other "addition" - is in reality a different function. This whole deal always revolved around how we write down stuff.

@mrpedrobraga 3 жыл бұрын

@@blinded6502 You can't add two vectors, addition is an operation between scalars and scalars, what you should do with vectors is addition. On that hand you can add vectors with vectors, but you can't add two scalars. Clearing that misunderstanding up :)

@luiscamarillo2322 2 жыл бұрын

I always had this problem with torque and angular momentum. I remember how after a Classical Mechanics class, I started searching for a more profound meaning to those concepts, since I was already in third year of Physics and still got no feeling of understanding the actual concept completely. Now I watched this video, and all of a sudden my soul can finally rest. I mean, even Feynman, when trying to explain the conservation of angular momentum, talked about how the area was preserved. It makes total sense.

@daniyelplainview 2 жыл бұрын

Torque, angular momentum, area, volume... all bi and tri-vectors. For regular vector algebra, it felt like, directionality is property that appears and disappears at will. Why the hell an area is a vector, volume is not? This makes perfect sense.

@dexter2392 2 жыл бұрын

@@daniyelplainview Yeah, volume isn't because trivectors in 3D are pseudoscalars.

@AkamiChannel 2 жыл бұрын

Can anyone link where Feynman talks about that (or tell me what to search bc links on youtube are usuallly a no-no)?

@nathanaelcase2783 Жыл бұрын

@AKAMI Probably “Feynman’s lost lecture” on Kepler’s laws .

@angeldude101 11 ай бұрын

After learning GA, I've started using the term "Kepler's Law of Quantum Mechanics" in a few places, because bivectors as areas (while not actually the most natural interpretation of them from what I've seen) trivially equates Kepler's Law to the Law of Conservation of Angular Momentum. Treating bivectors as areas also gives another interpretation behind the angle doubling: exp(aB) for a unit bivector B gives a rotor that covers a sector of the unit circle with an area of a. This way of viewing the bivector exponential extends to hyperbolic rotations and translations as well.

@milandavid7223 2 жыл бұрын

"Charge density is a current moving through time" Damn

@lukejames3039 2 жыл бұрын

LMAO

@pabloagsutinnavavieyra2308 2 жыл бұрын

That will definitely take me some time to process it in all its implications

@debblez 2 жыл бұрын

no, no, he’s got a point

@angeldude101 2 жыл бұрын

I've been looking at the Spacetime Algebra formulation for electromagnetics and it gets even crazier. When you add a separate basis vector for time, the charge-current is no longer a multivector with a vector part and scalar part. The charge-current in Spacetime Algebra is a pure vector, with charge being nothing more than the timelike component. Similarly, the electric field gains a timelike factor, turning the electromagnetic field into a single bivector with 6 components in 4D spacetime. Even with these modifications to the formulation, Maxwell's Equation still remains ∇F = J. Because WTF is up with this wizardry‽

@debblez 2 жыл бұрын

@@angeldude101 cool facts! thanks for sharing

@kunalverma6940 2 жыл бұрын

I swear I haven't had more surprises in 44 minutes in my entire life! This is one of the most underrated video on all of KZfaq.

@jamesperry5208 3 жыл бұрын

Just realised that the number of components of a K-vector is described by the Kth row of Pascal’s triangle. Beautiful

@housamkak646 Жыл бұрын

wowww whattt

@Michallote 2 жыл бұрын

I have noticed many people has had the same actual reaction that I had. I really cried, and I didn't understand why at first. In my case I believe it is because I experienced something so potent that it almost dwarves everything I know in terms of physics. I can't help but feel for something so beautiful, so deep that my soul finally rests on knowing that those doubts I had and my inability to sometimes grasp concepts is perhaps justified. I am almost finishing mechanical engineering and I really felt there was still too much to many knowledge I lacked, not because of not knowing the existence of such topics but feeling I wasn't on top of many subjects to the degree my curiosity demands. This was only mentioned briefly but the connection between tensors and complex numbers was something not even in my wildest dreams I could have come up and here it is JUST SO NATURAL. I always felt there was something missing from my education and I have looked desperately everywhere. I can't even begin to explain how relieved I am to finally arrive at this, not because it has solved all of my doubts, but because it gives me hope that I can perhaps eventually solve them. It's a light on a path I thought was shoruded in darkness. I just want to scream, we should all have been introduced to this as kids. And I believe all of the physics education programme should be built around it. It's something you can immediately and intuitively relate to.

@julesk1088 Жыл бұрын

♡♡♡

@haniamritdas4725 Жыл бұрын

Well said. But the reason that the subjects are not taught to everyone isn't that they are too hard. They are kept difficult precisely because the powers that be are not interested in a large population of mathematically capable individuals who are not under surveillance by institutions of government, education, or corporation. National security is why we cannot have nice things.

@DavidSartor0 9 ай бұрын

This video is not a part of your "Favorites" playlist. Why is this?

@sietsebuijsman8523 2 жыл бұрын

I'm completely amazed. I found a Wikipedia article on Geometric Algebra a while ago, seemed interesting, but I didn't quite get it. The past few days I've been working with quaternions and now your video comes along. I almost cried when the "i" entered the stage, it's all so incredibly natural. Thanks for blowing my mind.

@l0_0l45 2 жыл бұрын

Same!

@paulwolf3302 2 жыл бұрын

I never saw any explanation of i before that made sense. ijk=-1 made even less sense (quaternions). This math should be used instead of what we were all taught.

@animowany111 2 жыл бұрын

I actually paused the video to laugh for a good minute when 'i' appeared in the 2D case, then again when the comparison to quaternions appeared, then several times again for Maxwell's equation(s). For some reason this is funnier than comedy.

@hernanposnansky4830 2 жыл бұрын

Sietse Have got to the point of expressing the rate of change of unitary quaternions to the angular velocity pseudo vector of the rotating body? Essential for attitude determination.

@angeldude101 2 жыл бұрын

@@animowany111 For me, comedy is when you really should've seen something coming, but didn't. Seeing familiar elements pop up unexpectedly when doing something completely different and not noticing at first definitely counts as comedy in my eyes.

@andrew7955 Жыл бұрын

I love that Maxwell's equation is basically the same as the tensor notion, which also condenses the 4 equations down to 1. It would be cool to see how geometric algebra relates to tensors

@davidpalomino9138 Жыл бұрын

bivectors are also antisymmetric tensors

@codetoil 11 ай бұрын

Multivectors are antisymmetric tensors.

@matthewfreudenrich6557 2 жыл бұрын

I literally shouted when spinors came up out of no where. This is an absolutely amazing algebra and I am excited to see more of it.

@michaelpolioudakis 2 жыл бұрын

Wait a minute ,that double turn reminds me off something ...?!daaaamn...quantum mechanics...

@ixion2001kx76 2 жыл бұрын

If you like that, try Geometric Calculus.

@pedrovillanueva6767 2 жыл бұрын

Lmao me too

@julesk1088 Жыл бұрын

@@ixion2001kx76 Oh my gosh are we about to learn in a natural way why derivatives, integrals, and infinitesimals work

@TheGroundskeeper 11 ай бұрын

It should be noted that Pascal’s Triangle seems to tell how many of each component there are; in 3D it has 1-3-3-1 scalar-vector-bivector-trivector terms. I find this interesting

@azmah1999 9 ай бұрын

It's easy to prove this connection. Have a go at it !

@smolboi9659 4 ай бұрын

Yea the link is combinatorics. E.g we have 3 choose 2 ways to get our bivectors in 3D.

@juliavixen176 2 ай бұрын

It's the same pattern when multiplying polynomial equations of the same order.

@cmilkau 2 жыл бұрын

37:00 it actually IS the traditional gradient if you use relativistic coordinates (ct,x,y,z)

@valeriobertoncello1809 2 жыл бұрын

plus the identification of scalars and vectors pointing in the time direction

@flamingowrangler 22 күн бұрын

the d'lambertian, right?

@smorcrux426 2 жыл бұрын

this feels like a video that would be unimaginably useful in the future, but now I just lost it around the 20 minute mark. It feels as if I will return to this in the future, and find it really helpful. Edit: I remembered this video again now 5 months later, and I feel like I understand most of it! I've massively improved in maths in the past few months, and I'm even starting a bachelor's degree in just a few weeks.

@julesk1088 Жыл бұрын

How's the bachelor's going :)

@smorcrux426 Жыл бұрын

@@julesk1088 nice! I got 89 on linear algebra and I'm right now doing set theory

@eldattackkrossa9886 Жыл бұрын

@@smorcrux426 good job :)

@smorcrux426 Жыл бұрын

@@eldattackkrossa9886 thanks! My semester is starting in just a few weeks and this time I'm taking some computer science courses and real analysis

@krishnadasnair8765 Жыл бұрын

@@smorcrux426 hey dude just here to say best of luck for you courses!

@vulpesaxis8494 2 жыл бұрын

What? this is awesome, I'm about to finish my major in mathematical enginneering and not once have I heard of geometric algebra

@hawzhinblanca 3 жыл бұрын

please consider making part 2 with physics

@pedrovelazquez138 2 жыл бұрын

I love the fact that you used Tau instead of Pi... these are part of the essence of a revolution in math, where we need to make it shine in its intuitiveness.

@alberto3071 2 жыл бұрын

I don't usually leave a positive comment on a video, but I'm making an exception. This is a masterpiece, the knowledge to produce this must be so utter. I haven't seen a book or video about the topic more enlightening. My most sincere congratulations, you really have a gift for this, consider making a career out of it.

@kikivoorburg 2 жыл бұрын

Am I right in suggesting that this also perfectly explains why the cross product with Nabla/Del gives us a measure of Curl? It should actually be an outer product which would naturally results in bivectors, that are rotation objects! If I’ve got it right it’s absolutely gorgeous!!

@sudgylacmoe 2 жыл бұрын

Yep!

@geometrischealgebra1316 2 жыл бұрын

To quote William Kingdon Clifford from his founding article on geometric algebra: „ I may, perhaps, therefore be permitted to express my profound admiration of that extraordinary work, and my conviction that its principles will exercise a vast influence upon the future of mathematical science“ Clifford was refering to Graßmann, I would like to extend the compliment to your great video!!!

@rowantreeofknowledge1085 2 жыл бұрын

This video caused me to go on an unhinged rant to my non-math friends, which is always a fun time. This genuinely blew my mind over and over, and was fascinating and informative. Thank you so much for making this, and I pray to God you make more videos like it

@DiracComb.7585 2 жыл бұрын

KZfaq wouldn’t shut up with recommending me this video. And now I know why. This is a freaking gold mine. Nice work 👍

@TheOriginalJub 2 ай бұрын

I.... My mind is effin' blown. I'm trying to finish my PhD in mechanical engineering this semester, at MIT. This is my 26th semester at MIT, counting previous degrees. And somehow, I had never even heard of geometric algebra until this week. Suddenly, the math I was stuck on for the last two weeks seems trivial. THANK YOU

@mami42g 3 жыл бұрын

This video is an instant classic. I'll definitely revisit this later and I hope you create more mathematics/physics related content. Got a new sub :)

@durul-ny6ig 2 жыл бұрын

kac yasindasin

@trejohnson7677 2 жыл бұрын

Ehh

@ramansb8924 2 жыл бұрын

He man checked your playlist on mathematics really awesome ✨✨✨

@mami42g 2 жыл бұрын

@@ramansb8924 I didn't think anyone would see that :) Glad you liked it!

@ramansb8924 2 жыл бұрын

@@mami42g 🙌✨

@cmilkau 2 жыл бұрын

38:00 a current that is moving through time instead if space Yes! This intuition is really missing from so many textbooks and it really makes relativity so much more obvious. Density is a flow in time!

@williejohnson5172 2 жыл бұрын

Density is negative mass

@DavidLalo 2 жыл бұрын

This is the best math video in KZfaq. Every single "wait a second" had me feeling shock and joy. Everything just falls out so naturally!

@ricardowilsonaguiardacruz4554 2 жыл бұрын

I'd like very much to see treating the Navier-Stokes full equation set (mass, momentum and energy) by this tool. The tensor treatment of those equations is too tedious.

@paulwolf3302 2 жыл бұрын

Tensors are just a headache.

@hernanposnansky4830 2 жыл бұрын

Ricardo good idea but start with the Euler equations (zero vicosity)

@miroslavjosipovic5014 3 жыл бұрын

Nice presentation. Some remarks as support: 1. Note that multiplication rules for orthonormal vectors (at 14:40) are the same as the rules for the Pauli matrices, which gives a direct connection to quantum mechanics. The Pauli matrices are just a representation of Cl3 (geometric algebra of 3D Euclidean vector space). 2. We can express classical mechanics in Cl3, calculations are much faster (up to 10 times), the Kepler problem can be reduced to a harmonic oscillator problem, we do not need tensors (like inertia tensor), etc. Then, we can use the same mathematics (Cl3) for EM theory (as in this video), the special theory of relativity (we do not need the Minkowski 4D spaces), quantum mechanics (without the imaginary unit), Dirac theory, even the general theory of relativity. Briefly, all main theories of physics with the same language, without imaginary numbers, matrices, and tensors. 3. As for the possibility of generalization and unification, just two examples. a) Once you write down a rotor formula in 3D, you can apply it immediately in, say, 121D. b) In geometric algebra, all integral theorems of the vector calculus (including those from complex analysis) are united in just one theorem (the fundamental theorem of calculus). 4. Geometric algebra likes programming; we almost do not need "if-then-else". 5. Without matrices, but with oriented numbers (k-vectors), geometric algebra gives us a geometric clarity at each step of calculations. This is very important, because it employs the tremendous visualization power that human beings have. 6. Geometric algebra sheds new light on the concept of numbers (oriented numbers). Note that Cl3 contains real numbers, complex numbers (but without the imaginary unit), hyper-complex numbers (say, vectors), dual numbers, quaternions (the even part of Cl3), spinors (they are just quaternions here), etc. See also www.springer.com/gp/book/9783030017552 (a book recommended by Hestenes), www.researchgate.net/publication/343382199_Dirac_theory_in_Euclidean_3D_Geometric_algebra_Cl3

@ChaineYTXF 3 жыл бұрын

Hello, thank you for the two references. Do you have any reference for in depth explanation of point 2: expressing Classical Mechanics in _Cl_ 3? Thank you in advance, or thanks to anyone contributing to the answer.

@miroslavjosipovic5014 3 жыл бұрын

@@ChaineYTXF Hestenes: New Foundations for Classical Mechanics

@ChaineYTXF 3 жыл бұрын

@@miroslavjosipovic5014 thank you very much. Hestenes seems at the heart of the debate when discussing geometric algebra. I'll look it up.

@miroslavjosipovic5014 3 жыл бұрын

@@ChaineYTXF Hestenes revived GA.

@uhbayhue 3 жыл бұрын

This video blew my mind, again and again and again. Thank you so much for this beautiful presentation

@mickwilson99 2 жыл бұрын

I so wish I'd had explanations like this when studying physics 40 years ago! The intuitions are rendered apparent and the "just because" statements eliminated.

@Rh0mbus 9 ай бұрын

This video genuinely feels life changing to me. This is like you have finally given access to all the colors as an artist. This is absolutely gorgeous... I'm stunned

@mathunt1130 2 жыл бұрын

I did a postdoc in geometric algebra applications to 5 axis machine milling. We used a version of geometric algebra which no one other than two people in the world use.

@mrdr4934 Жыл бұрын

I am a nerd,I love science and math, I love learning about things even when I don't need to. I've seen a lot of math videos because of this. I have never felt so invested into a price of media in my entire life. This video has single handed changed my view of geometry. This is a masterpiece

@DavidSartor0 9 ай бұрын

Subscribed.

@rhke6789 2 ай бұрын

Best set of videos on the algebras I have thus far viewed. congrats. Very easy to understand and presented clearly. Viewer like me feel rewarded after viewing and very grateful

@elliotwaite 3 жыл бұрын

This seems like the future of understanding physics. I'm hoping you make more videos about geometric algebra 🤞. Your ability to explain the subject matter has been the best I've found so far. And your use of manim is top notch.

@williamnelson4968 3 жыл бұрын

I finally "Grokked" Geometric Algebra thanks to your visual presentation. I don't even think 3Blue1Brown could enlighten Geometric Algebra this well. Many thanks!

@bonusgeese6860 2 жыл бұрын

Lol i literally thought that this was a 3blue1brown video until I found this comment (I don’t recognize voices very well)

@hierkonnteihrewerbungstehe5636 2 жыл бұрын

@@depression_isnt_real Don't get me wrong: 3b1b videos are amazing, but what's even better is that he made the tool he created for creating those videos open source. I can't wait to see how many great math videos in the 3b1b style like this one will be out there in a few years created by people like you, inspired by 3b1b. I believe that this is 3b1b's biggest gift, even bigger than his videos.

@evanev7 2 жыл бұрын

Yeah, I had an idea about it but this really sold it for me

@Octavarium666xyz 2 жыл бұрын

@@depression_isnt_real And few years ago it was "Sal Khan's black backdrop with hand written notes" type of videos.

@alejrandom6592 2 жыл бұрын

no need to compare

@ElZafro_ 2 жыл бұрын

I can't even express with words how mindblown this video left me. Amazing work, the concepts made sense so easily.

@jai_nasu 22 сағат бұрын

At 38:00 when you said charge is current moving in time, my brain imploded with the spacetime diagram and worldlines on it. Thank you.

@SeanGhaeli Жыл бұрын

I am totally amazed by this. It was during quantum mechanics when I was thinking to myself that physics just gets too complicated and abstract for my brain to keep up, but in 45 minutes you've shown that with the right language, all natural phenomenon can be expressed in a shockingly simple way. When you pulled out the pauli spin matrices, I was caught totally off guard.

@michaeljburt Жыл бұрын

Master's in electrical engineering here. Absolutely floored. This is a great distillation of a topic that I have been always wary about touching. I am eager to step through this reformulation of electromagnetics to see what I might gain in the new viewpoint. Excellent!

@ITAlife98 2 жыл бұрын

This might be the best math video I've seen on KZfaq. Suddenly everything about vectors and complex numbers just clicks and makes perfect sense. I always had a hard time conceptually understanding complex numbers, cross products etc., I was always taught about them in a disjointed way. Now I see it's all a beautiful interconnected web.

@LunizIsGlacey Жыл бұрын

Thank you for the corrections in the description. As someone who's into abstract algebra, some things in the video struck me as strange but intuitive, so I let them slide so that I could get an intuitive sense of what you were discussing. But after the intuitive comes the specifics - and pointing out where intuition goes wrong. And this is exactly what you did in the description! Thank you for the very well-taught broad and intuitive overview of geometric algebra, it seems very fascinating. And for the specificity and footnotes in the description!

@kurtfeuerstein1907 2 жыл бұрын

Looks like a proper formalism to describe both, field theories and quantum mechanics. And it's so easy and illustrative that it even could be taught at high school.

@johanvandercingel 6 ай бұрын

What an immensely clear presentation of such elegant concepts. This brings a lot of new concepts to my mind that is littered with inconsistent school book ideas that did not make sense. Now you have connected the dots for me and I see a picture that brings this together. THANK YOU!!! Please bring more of this to the world.

@fernandogarciacortez4911 2 жыл бұрын

Man, for some reason I shed a tear at the end. What a great video, thanks a LOT for sharing. As an undergrad in physics, this just adds even more motivation to pursue more knowledge. Such a beautiful topic, I will definitely read more on it.

@addy7138 3 жыл бұрын

I am in High school when you explained bivector are equal to imaginary number, I screamed in mixed emotions of horror and excitement and joy

@jelleverest Жыл бұрын

You've sent me on an absolutely wild goose chase. I've found lectures by David Hestenes and worked through his books (or at least tried). Thank you for introducing this!

@ShayminMaster2 Жыл бұрын

This is by far the most impactful math video I've seen all year. I bought a book for game engine development and it touched on Clifford Algebra, but this video brings everything together. Thank you so much!

@superscript9367 2 жыл бұрын

You're an incredible teacher. The presentation of this is brilliant and so well laid out! It somehow managed to keep my attention for the entire 45 minutes. I'm a physicist and never had to learn this formalism, but now that I know it it is so useful. Thank you!

@Siluetae 6 ай бұрын

Thanks for the assist... I've been working on this for awhile, because I knew it had to exist! Incredible, we can ALL breath now and refocus, the chains are off now friends...

@IshayuG 9 ай бұрын

At 22 minutes in I've seen every single formula before and had no idea they were so related. You are a teaching wizard.

@hughhanginloose Жыл бұрын

This video was incredible; truly a service to humanity. I hope someday we start teaching these concepts to much younger students so they can absorb the unity of geometry, algebra, and physics while their minds are still plastic. It really makes all obscure concepts that students struggle with so much more intuitive! You've changed the way I think with this video, I can't thank you enough. I wish you nothing but good things :)

@alexanderherbertkurz 3 жыл бұрын

I have been looking at Geometric Algebra for a couple of days now and this is by far the best first introduction I have seen so far. Thanks!

@VMac-eg7fb 2 жыл бұрын

Your teaching style carried me a novice through your presentation, didn't understand hardly anything but somehow had enough grasp to eagerly finish without boredom enjoying every word. Such a powerful tool. Thank you for clarity of concepts, 79 yrs senior, non math major.

@halewen7240 3 ай бұрын

i major physics, and recently been wanting to study some math. My math nerd friend suggested me to start from geomtric algebra, and i came across this video. Almost had heart attack when the scalar adding vector shows up, but after that, every his "wait a minute" blows my mind. This is such a great video! and I could never view Maxwell equation the same way again.

@alejrandom6592 2 жыл бұрын

This is so amazing!!! I have a more or less basic knowledge on vectors, matrices, complex numbers and quaternions and never heard about geometric algebra, this is beautiful!

@davidhidary Жыл бұрын

Absolutely mind blowing, this has fundamentally altered my perspective of physics math and how they relate. Please keep doing what your doing, maybe even delve into some of these topics like the Pauli matrices ect.

@robinwang6399 Жыл бұрын

My mind is blown. Time to extend what I learned with this. Edit: why is this not taught in my physics program? This is so important!

@kushlord2139 Жыл бұрын

You blew my mind when you connected this to complex numbers at ~20 minutes in. Finding connections in mathematics is one of my favorite things!

@ZedaZ80 2 жыл бұрын

This is a damn good video! I had to pause, go back 20 seconds, and rewatch several times, and each time I would end up understanding what was going on. I'll be revisiting this video for sure; you did an amazing job.

@potatofries571 2 жыл бұрын

Absolute banger of a video my guy Definitely unified many concepts I’ve studied in my physics degree and in a super concise/simple/EFFICIENT way!

@pabloagsutinnavavieyra2308 2 жыл бұрын

In my second course of quantum mechanics we saw sponsors and I found odd their properties and tried to reconcile their geometric interpretations as I did with complex numbers and quaternions. This has brought me a grant relief and also now I'm excited to look into Pauli Matrices as well. Thanks for making this video. I tried reading on Clifford Algebras before, but I was having complications following some things and couldn't get to the juicy parts. But with the animations and when you mention at the begging that just having a simple notion of why they are helpful on the first watch was enough really helped to put me in a swift moving mindset to absorb almost all what you said (while making some pauses to ponder a little on some ideas I explored before on my own). I bet making this video took a long time to make, and I'm so grateful that you take the time to do so. Thank you so much!

@adamcummings20 6 ай бұрын

My god. I remember watching this video 2 years ago and thinking "hey thats pretty cool". Now I'm doing a masters degree in mathematics, trying to wrap my head around differential geometry, and seeing the way you've explained the wedge product has blown my mind, and helped me through this absolute slog of a degree 😂. Many thanks for this amazing video

@PenguinMaths 3 жыл бұрын

This video is amazing quality, thanks for making it! I hope to see you make more videos like it

@maxwellsequation4887 2 жыл бұрын

Your videos are also amazing!

@paulwolf3302 2 жыл бұрын

I feel like my idea of vectors has been expanded like a balloon.

@Headbangnuker 3 жыл бұрын

Brilliant. Having ventured into Manim myself, the effort in this shows, thank you.

@ultramadscientist Жыл бұрын

This whole video was such an "oh shit" moment you laid it out so perfectly I was like "Oh!!!! I know I know I recognize that that's SPINORS" and you're like "Hey wait a minute that's spinors" but for literally every example. I feel like I finally understand

@nothinginteresting1662 Жыл бұрын

Notice that in n dimensions, the multivector has 2^n components. 2 in 1D -> 1 scalar, 1 vector 4 in 2D -> 1 scalar, 2 vectors, 1 bivector 8 in 2D -> 1 scalar, 3 vectors, 3 bivectors, 1 trivector Do you see a pattern? Binomial theorem coefficients. The sum of all coefficients is always a power of 2 and in n-dimensions, the multivector has nCr r-vectors. r=0 is a scalar, r=1 is vector, r=2 is bivector and so on. So for 4D, there will be 4C0 = 1 scalar, 4C1 = 4 vectors, 4C2 = 6 bivectors, 4C3 = 4 trivectors and 4C4 = 1 quadvector (or 4-vector), total 16 components, which is 2^4

@physicsmathematics7445 Жыл бұрын

This is the best thing ever happened. I literally watched entire video with goose bumps. I desperately needed some strong unification of electrodynamics under vector calculus and this has just made my life probably. Thank you so much 💓

@morganhopkins204 3 жыл бұрын

I like to think of blades geometrically by extending the geometric description of vectors. We identify a vector *v* by its magnitude and direction. That is, the geometric properties | *v* |, Span{ *v* }, and a choice of orientation (+/-) determine *v* . In this view, a vector is an oriented quantity associated with a particular linear subspace. This is readily extended to all k-blades B = *v1* ^ ... ^ *vk* as an oriented quantity ±|B| associated with a linear subspace Span{B} = Span{ *v1* , ..., *vk* }. I feel like this perspective is more conceptually flexible, as blades are used to represent a variety of geometric and physical relationships, such as areas/volumes/etc., rotations, torques, and even linear subspaces themselves. Thinking of blades generally as quantities associated with a subspace, rather than something more specific like an oriented area/volume/parallelepiped/etc., helps unify these seemingly disparate ideas.

@drdca8263 2 жыл бұрын

@@hyperduality2838 stop spamming this duality junk everywhere, it isn’t actually insightful. There are things about dualities that can be said and which are insightful, but none of the things I’ve seen you say about (alleged) dualities are at all insightful, and the ones that aren’t trivial are generally wrong. If you want to say some somewhat more interesting things about dualities, learn some category theory at least to the point where you understand the precise sense in which products and coproducts are dual concepts. There are valuable things to say about dualities, but the stuff you have been saying just sounds like someone who maybe did some recreational drugs and mistakenly concluded that they gained some great insight.

@MahdiEsmaily 6 ай бұрын

Hands down, this has been perhaps the best youtube video I have seen in the past 20 years. So many Ahhha moments! Beautifully explaining the original of many relationships that I took for granted or didn't understand where they come from.

@MountainCodeMonkey 2 жыл бұрын

I have read up on these topics for years and never really understood them fully. This was the most informative thing I've ever encountered on any of these topics! WELL DONE!

@rubetz528 2 жыл бұрын

Man, you deserve way more subscribers than you have. I hope you'll keep these videos coming. It was really informative and very insightful. I was really delighted after I had thought "why need outer product when you have cross product?" and you addressed it some 5 minutes later. Besides, it was really cool how you pointed out that physicists should get more familiar with geometric algebra as I can see a significant need for a more advanced mathematical apparatus in physics too. Many kudos to you! You're a guy I would buy a beer IRL!

@overlordprincekhan 3 ай бұрын

Ayoo bro, you just made sense to my 20 years of education in just one video. WTF! IDK how tf everything makes sense now

@HebrewJones 2 жыл бұрын

Wow, I watch the whole video three times just to grasp all the concepts you’ve presented and it just opened my mind, thank you, other math videos just confirms things I already knew you made me think

@julianociaramello2150 Жыл бұрын

... you've got to be kidding me... this would have simplified just... so damn much... Incredible.

@mark11tz 2 жыл бұрын

I am a physics major and I sincerely thank you for your great work on this video. You solve my confusion lasting for many years when I encounter with multi-dimensional calculus. Although sometimes we just get used to the "rules" we maneuver the terms, I somehow feel uncomfortable to the interpretation of the equations. The concept of geometric algebra you introduced to me just take off the frame which trapping my mind for so long. I wish I could see this video earlier.

@rayujohnson1302 Жыл бұрын

Thanks for this. I have a bachelors in electrical engineering so I was familiar with some of this, but this video brings it all together beautifully in a way that makes me wonder why it isn't taught in college this way.

@Dinhjason 2 жыл бұрын

The explanation of imaginary numbers being pseudo-scalars was a clear Eureka moment, this video was a genius explanation!

@billy-cg1qq Жыл бұрын

All my physics classes and struggles flashed before my eyes lamenting how simple could it be if a knew this elegant and beautiful representation of geometric algebra.

@asdf56790 Жыл бұрын

Thank you SOOOOO much for this video! I'm currently taking a differential geometry course and this video gave me insanely valuable intuitions for working with differential forms!

@_wetwillyinc 2 жыл бұрын

16:21 "The outer product is a new equation though, or is it??" then he shows the cross product wooowwoowowowowo my head

@_wetwillyinc 2 жыл бұрын

and then he just calls ^x^y the imaginary unit and proceeds to rotate vectors with it YOOOOOOOO 19:44 i^2 = -1 ohhhh we're in it DEEP now

@_wetwillyinc 2 жыл бұрын

23:53 if you thought the crazyness was over please, sir, thats all i have, i beg of you

@_wetwillyinc 2 жыл бұрын

26:31 aaaaand there goes the righthand rule THERE IT GOES

@_wetwillyinc 2 жыл бұрын

27:42 kronk: oh yeah its all coming together 28:43 alright i got out of my chair and started jumping this is probably the biggest awakening of my entire undergrad youtube studies im so hype WOW

@_wetwillyinc 2 жыл бұрын

ITS QUATERNIONS TOO!!!!!!! ITS QUATS BRAH HOW

@stevelam5898 2 жыл бұрын

That must be the best mathematics lecture I've ever watched in my entire life: complex analysis, geometric algebras and the derivation of Green's & Stokes equations made simple in less than 45 minutes.

@BHBima Жыл бұрын

First time learning geometric algebra and I love it. everytime you say "wait a minute" it makes me smile knowing that it's going to blow my mind.

@davidhuo6902 2 жыл бұрын

It is just the missing lesson in the university, which I wanted to know for years, connecting linear algebra to higher level math concepts and dozens of physics theories.

@hawzhinblanca 3 жыл бұрын

i was hoping someone would do a video like that, very helpful, thanks for making it

@AmatsukiHane 11 ай бұрын

As a Physics student my mind was blown again and again during this video... Knowing a mathematical object that connects so many physical concepts (pseudo scalar, cross product, Pauli matrices, spinor...) that seem to be unrelated is just amazing

@LydellAaron 2 жыл бұрын

Tremendous video! You can really start to see the mathematical foundations with this geometric algebra. Thank you for taking the time to put this together for us. You have even specifically helped me realize a definition I would use to describe/model "entanglement" in a quantum computer at 16:21 and on through 16:25 which is the multiplying out of cross-component differences between two vectors in this case.

@eneserdogan34 2 жыл бұрын

learning geometric algebra feels like a learning different language (or programming language). Completely different mindset.

@cosmos821 2 жыл бұрын

OMG! Geometric Algebra just became my favorite math subject. Please keep making videos!!!

@michaelwish7271 Жыл бұрын

I am finishing a dual math/physics masters and I can't believe I've never seen this. My head exploded when the bivector/imaginary number connection was made. I love physics, but math never ceases to amaze me.

@MiroslawHorbal Жыл бұрын

I've watched this a few times now and it just keeps blowing me away. Been playing around with the 2D case and it's pretty incredible how complex numbers just fall out as a requirement. It also is pretty wild that the interpretation of complex numbers as representing actions on a vector (scale and rotation) just comes naturally. Multiplying vectors gives you the action of rotating from one to the other and scaling by the magnitudes. Conversely, given a vector and a complex number, you can solve to find which vector multiple will generate that complex number action. There's a dual universe between complex numbers and vectors in a 2d vector space just beneath the surface.